1 Introduction

Numerous practical applications entail the optimization of specific goals such as energy conservation, environmental protection and performance, efficiency, and long-term viability (Yang 2020). The optimization problems which can be framed in several circumstances are very complex, with multimodal objective landscapes and a collection of complicated, nonlinear constraints (Yang 2020). Solving such difficulties is challenging. Using basic brute force tactics remains unrealistic and undesirable, despite the rising capability of modern computers. As a result, efficient algorithms are critical in such applications wherever feasible. Unfortunately, efficient techniques may not present again for majority of application optimization problems. MLT-based image segmentation is an example of such problem in which a proficient search of the solutions inside a complex search area is required to discover the best solution (Dhal et al. 2020a). MLT is useful for generating two or more homogenous classes by segmenting composite images. The main procedure in the field of MLT is to determine the best threshold levels. Nevertheless, the MLT techniques have a major drawback in terms of the computation time that basically tends to increase as the number of thresholds grows, making it computationally demanding. Such flaws of MLT can be resolved by the Nature-Inspired Optimization Algorithms (NIOA) and it has become an imperative possibility. This inspiring nature has triggered the inquisitiveness in many academic scholars thereby focusing on the development of NIOA by conceptualizing natural events in computational terms. NIOA and their improved variants have proved its efficacy in engineering optimization problem and also resolving several MLT problems. Therefore, this study concentrates to accomplish an up-to-date review on recent NIOA and their MLT applications during past three years i.e., 2019–2021.

The remaining sections of the paper are systematized as follows: The review and discussion on recent NIOA since 2019 is presented in Sect. 2. The application of NIOA over MLT problem has been reported in Sect. 3. Section 4 highlights the experimental results of some newly developed NIOA in the year 2021. Lastly, Sect. 5 discusses on the conclusion and the future research directions.

2 Survey on recent nature-inspired optimization algorithms

Since millions of years, flora and fauna have showcased mechanisms to protect and secure their survivability when sources are scant (Li et al. 2020a). It's realistic to opt these tactics for the prevalence and effectiveness while developing optimization algorithms. NIOA algorithms have offered a multitude of meta-heuristic algorithms for solving anticipated issues for the last few decades. According to the No Free Lunch (NFL) theorem, there exist no single NIOA that has the capability of solving all optimization problems (Dhal et al. 2019a, 2020a; Wolpert and Macready 1997). As a consequence, scholars are constantly proposing new NIOA, preserving the sector and propelling substantial progress year after year. In the work of Dhal et al. 2020a, the list of NIOA and their application in MLT domain has been reported up to the year 2018. Houssein et. al. (Houssein et al. 2021a) published a book chapter in 2021 in the same topic but their research does not contain the recent full list of NIOA. The complete list of the developed NIOA in the period of 2019–2021 is reported in Table 1. In recent times, the combinatorial optimization field has perceived an overflow with respect to "new" NIOA, the majority of which are based on some natural or artificial metaphor. Therefore, one only question that arises in the mind of the researchers when working in this field i.e., “which NIOA needs to be selected and focused on for the problem?”. Therefore, the Table 1 also highlights the citation of the NIOA because citation can be a good metric for selecting NIOA from this huge set. It can be observed that in the year 2019, 2020 and 2021, around 22, 25, and 16 different NIOA respectively have been developed and proposed world-wide. However, such speedy progress and introduction of new NIOA makes the research tough in this field. Sorensen (Sörensen 2015) also wrote in his paper that this massive numbers of algorithm drags the field of NIOA a step backward rather than forward. His research also gained enormous popularity thereby achieving 726 citations according to Google scholar (dated 26.10.2021). As a consequence, the most critical question that arises in the mind of many today is “Does the scientific community need new NIOA’s or the existing NIOA’s are enough?”. There are already enough "novel" NIOA, according to the author of Sörensen (2015) and Fister et al. (2016), and there is no need to introduce additional NIOA. The time has arrived for standardization, to permit the research society to emphasis on more promising research avenues in the NIOA literature and to identify the true mechanics underlying in these "new" NIOA. The year wise number of published papers over NIOA based MLT has been presented in Fig. 1a.

Table 1 List of recently introduced NIOA
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Fig. 1
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a Year wise published papers over NIOA based MLT; b General Flowchart of the NIOA

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On the other hand, no matter how rapid and massive is the evolution of Nature Inspired algorithm, NIOA is the scorching research area that has not just attracted several researchers but proved to be the most rapid growing field in research. Therefore, the introduction and designing of an integrated framework in terms of algorithm structure is advantageous for implementation. Over the last few years, a large body of literature has offered priceless insight into how to comprehend the applied tactics and what the NIOA's general characteristics are (Dhal et al. 2019a, 2020a, b, 2021a). Based on the same, the common procedure of the NIOA is presented using flowchart depicted in Fig. 1b.

Most of NIOA follows the common flow comprising of four major steps as described in Fig. 1b. In Step 1, the population and its related initial parameters, representing possible solutions to the given optimization problem is initialized. Typically, random methods are applied to generate the initial parameter of the population ensuring that it conceals the solution space as much as possible. Based on the specific requirements of the experts and their past experience, the size of population (should be perhaps large) is selected. After initialization, in subsequent Step 2, the calculation in terms of the fitness value of an individual solution of population is performed and the concrete iteration loop starts, where every loop signifies a generation (Number of iterations). The fitness function which is most commonly considered as a unique pointer to echo the performance of each solution is premeditated by the help of the target function either comprising of the maximum value or minimum value. Commonly, each population has its own local optimal solution, and on the other hand, global optimum is owned by its complete population. The fitness values of the population are further computed in each of the iteration. Given the termination condition, if the global best solution satisfies the same, the output is generated (as shown in Step 4) else, Step 3 is instigated, that generally anticipates to accomplish the significant operations that is basically carried out for the purpose of information exchange amongst entire population to evolve excellent individuals. Further, updating in terms of population is carried thereby initiating the workflow to supplementary navigate towards Step 2 to accomplish the subsequent iteration. The generation counter is further amplified and till the stopping criteria is satisfied, the iteration undergoes. Stopping condition is a parameter that specifies when an algorithm should halt. For various NIOA, determining the stopping condition is critical. Number of Iterations, commonly known as NIs and number of Function Evaluations frequently known as FEs are two of the most prevalent types of stopping criteria. According to recent surveys, researchers prefer FEs to NIs (Dhal et al. 2020b).

2.1 Major challenges and problems with NIOA

Though there are immense number of researches and breakthroughs in regard to NIOA, it still has quite a few noteworthy challenges which can be pointed out by the following set of questions, especially from theoretical perspectives (Yang 2020):

  1. 1.

    Question on Mathematical Framework: How can a uniform mathematical framework be created so as to learn much about their convergence, convergence rate, consistency, and reliability? In reality, the core progressions of how such algorithms function, why it functions and under what circumstances are yet unknown and still mysterious.

  2. 2.

    Question on setting of NIOA’s parameters optimally: How to set the parameters of the NIOA to achieve efficient outcome for a set of problems? How can these parameters be meticulously changed or controlled to improve an algorithm's performance?

  3. 3.

    Question on importance of Benchmarking and NFL Theorem: The usage of benchmark functions to examine the performance of the new NIOA in contrast to other existing algorithms is of utmost importance. On the other hand, the NFL theorem asserts that if algorithm A outperforms algorithm B in achieving the best optimal values of certain objective functions, then B will outperform A on other functions. However, from experience the interpretation can be made that some algorithms are superior to others. So, how to reconcile these seeming inconsistency? Therefore, the questions are: What are different varieties of benchmarking that turns out to beneficial? Free lunches do exist or not? If so, what could be the available circumstances?

  4. 4.

    Question on Performance Measures: The present literature primarily focuses on accuracy, computing effort, stability, and success rates as the performance indicators. In general, Friedman test and the Wilcoxon’s rank sum test which are the widely accepted non-parametric tests can be used to compare and analyze efficiency among NIOA (Dhal et al. 2020b). However, it is unclear whether the aforementioned performance indicators are genuinely fair in comparison. Therefore, the questions are: What performance metrics can be considered in order to compare all the existing algorithms fairly? Is there any kind of possibilities to formalize a sole framework that allows entirely available algorithms to be related impartially and objectively?

  5. 5.

    Questions on Algorithm Scalability: The most crucial indicator of an algorithm's efficacy from the standpoint of application is how well it can tackle a wide range of issues. Therefore, the question is: How can algorithms that work effectively for small-scale problems is scaled up to efficiently handle truly large-scale, real-world challenges?

Apart from the above-mentioned issues, other ongoing difficulties with NIOA includes finding the best combination of exploitation and exploration, dealing successfully with nonlinear restraints, and applying these algorithms (NIOA) to machine learning techniques especially deep learning.

3 Recent trends in multi-level thresholding using nature-inspired optimization algorithms

It is reported in literature that when two or more segmentation approaches work together, they can yield better results than if they worked alone (Dhal et al. 2020a; Khan 2013). Multi-thresholding is computationally expensive combinatorial task that entail large and complex search spaces. The evaluation of all feasible outcomes leads to a time-consuming search process. As a result, researchers incorporated NIOA, a powerful search tool, in this domain to make it less complicated and fast. This section gives a brief mathematical formulation of MLT problem; general algorithm of the NIOA based MLT, literature survey on MLT since 2019, and the objectives functions which play crucial role for proper image segmentation.

3.1 Introduction to multi-level thresholding

The central concept behind multi-level thresholding is to identify threshold values which allow the segmented images to fulfill the required criteria. This would be accomplished by optimizing specific objective function/s, with the threshold values as input parameters (Dhal et al. 2020a).

Assume that the image f comprising of L gray levels needs to be segmented into p partitions \(\left({C}_{1}, {C}_{2}, \dots ,{C}_{i}, \dots {C}_{p}\right)\) using set of (p − 1) threshold values \(TH=\left({t}_{1},{t}_{2},\dots ,{t}_{i},..., {t}_{p-1}\right)\), where \({t}_{1}<{t}_{2}<,\dots ., {<t}_{p-1}\). For example, L = 256 for an 8-bit image and the grey levels are between 0 and 255 (Dhal et al. 2020a). Hence, a pixel containing certain gray level \(g\) belongs to class \({C}_{i}\) if \({t}_{i-1}<g<{t}_{i}\) for \(i=1, 2,\dots ,p\).The technique of determining the set of optimal thresholds \({TH}^{opt}\) that optimizes the objective function \(F\left(TH\right)\) is referred to as single objective thresholding. The mathematical expression is as follows:

$${TH}^{opt}=\underset{0\le \mathit{TH}\le L-1}{\mathrm{argmax}/\mathrm{min}}\left\{F\left(TH\right)\right\}.$$
(1)

For multi objective MLT, \(F\left(TH\right)=\left({F}_{1}\left(TH\right), {F}_{2}\left(TH\right), \dots ,{F}_{j}\left(TH\right),\dots ,{F}_{n}\left(TH\right)\right)\), where \(n>1\).

3.2 Objective functions

In the theme or context of Multi-Level Thresholding-based image segmentation, objective functions are crucial. Few widely employed objective functions that have been presented in the existing literature. Objective functions in Table 2 categorized into some classes such as Variance based like Otsu, 1-Dimensional histogram-based entropies like Kapur’s entropy, 2-Dimensional histogram based objective functions, 3-dimensional objective functions, hybrid objective functions like Fuzzy-Tsalli’s entropy. Finding novel objective functions or the optimum objective functions for various image kinds is likewise a difficult task. Recently, some entropy based objective functions have been devised for MLT based image segmentation like t-entropy, and Masi entropy. According to the research, the segmentation efficacy of objective functions is strongly influenced by the issue domain, i.e., image type. Figure 2 depicts the number of papers Surveyed using different types of objective functions in the last three years recorded in Table 3.

Table 2 Different types of objective functions
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Fig. 2
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Number of papers surveyed using different types of objective functions

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Table 3 Literature reports on NIOA based multi-level thresholding
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From Fig. 2, it can be seen that Kapur’s entropy and Otsu maintain their popularity as objective function till date where some other new objective function have been developed. Now, a brief mathematical implementation of Kapur’s entropy has been given below for the help of the reader relate better to the Optimization being performed.

Kapur method (Ray et al. 2021) considers the combination of a probability distribution function of a given histogram with the concept of entropy. The principal idea behind this method, aims to find the best threshold combination that maximizes the entropy. Considering a binary class segmentation problem, the objective function for Kapur’s entropy is defined by:

$$f\left( {th} \right) = \max \left( {H_{1} (th) + H_{2} (th)} \right),$$
(2)

where the entropies of each class H1 and H2 are computed as follows:

$$H_{1} (th) = \sum\limits_{i = 1}^{th} {\frac{{Ph_{i} }}{{w_{1} \left( {th} \right)}}\ln \left( {\frac{{Ph_{i} }}{{w_{1} \left( {th} \right)}}} \right)} , \, H_{2} (th) = \sum\limits_{i = th + 1}^{L} {\frac{{Ph_{i} }}{{w_{2} \left( {th} \right)}}\ln \left( {\frac{{Ph_{i} }}{{w_{2} \left( {th} \right)}}} \right)} ,$$
(3)

where i represents the intensity of a given pixel such that \(\left( {0 \le i \le L - } \right)\); NP represents the total number of pixels; h indicates the image histogram; hi indicates the quantity of pixels having i intensity; and Phi is the probability distribution of the i-th level which is defined as:

$$Ph_{i} = \frac{{h_{i} }}{NP}, \, \sum\limits_{i = 1}^{NP} {Ph_{i} } = 1,$$
(4)

and the cumulative distribution function of each class is defined as:

$$w_{1} (th) = \sum\limits_{i = 1}^{th} {Ph_{i} } , \, w_{2} (th) = \sum\limits_{i = th + 1}^{L} {Ph_{i} } .$$
(5)

For multi-level image segmentation, Kapur method can be extended to incorporate multi thresholding capabilities. In such case, the image I is divided into K classes. Under such circumstances, Eq. (2) is redefined as:

$$f({\mathbf{th}}) = \max \left( {\sum\limits_{i = 1}^{K} {H_{i} } \left( {{\mathbf{th}}} \right)} \right),$$
(6)

where \({\mathbf{th}} = [th_{1} ,th_{2} ,...,th_{nt} ]\), represents the vector of threshold values and the entropy values, corresponding to each threshold value for a multi-level segmentation problem is calculates as:

$$\begin{gathered} H_{1} (th_{1} ) = \sum\limits_{i = 1}^{{th_{1} }} {\frac{{Ph_{i} }}{{w_{1} (th_{1} )}}\ln \left( {\frac{{Ph_{i} }}{{w_{1} (th_{1} )}}} \right)} \, \hfill \\ H_{2} (th_{2} ) = \sum\limits_{{i = th_{1} + 1}}^{{th_{2} }} {\frac{{Ph_{i} }}{{w_{2} (th_{2} )}}\ln \left( {\frac{{Ph_{i} }}{{w_{2} (th_{2} )}}} \right)} \hfill \\ \begin{array}{*{20}c} {} & {} & {...} & {} \\ \end{array} \begin{array}{*{20}c} {} & {...} & {} \\ \end{array} \begin{array}{*{20}c} { \, } & {...} & {} \\ \end{array} \hfill \\ H_{K} (th_{nt} ) = \sum\limits_{{i = th_{nt} + 1}}^{L} {\frac{{Ph_{i} }}{{w_{K} \left( {th_{nt} } \right)}}\ln \left( {\frac{{Ph_{i} }}{{w_{K} \left( {th_{nt} } \right)}}} \right)} . \hfill \\ \end{gathered}$$
(7)

The cumulative distribution function of each class is defined as:

$$w_{1} (th_{1} ) = \sum\limits_{i = 1}^{{th_{1} }} {Ph_{i} } , \, w_{2} (th_{2} ) = \sum\limits_{{i = th_{1} + 1}}^{{th_{2} }} {Ph_{i} } , \, w_{K} (th_{nt} ) = \sum\limits_{{i = th_{nt} + 1}}^{L} {Ph_{i} } .$$
(8)

3.3 Recent literature on MLT using NIOA

This section encompasses an informed literature assessment on the usage of NIOA to solve the MLT problem. The study takes into account works that were released between 2019 and 2021.The general approach of MLT using NIOA is summarized as Algorithm 1 comprising of various steps. In the initial step i.e., Step 1, the objective function is considered. In MLT, the ideal thresholds is attained by optimizing a certain objective function or maximizing various entropy such as Kapur’s entropy, between-class variance etc. Further, the initial parameters of the population that essentially represents the solution to the given optimization problem are haphazardly initialized and created in the solution space. The fitness function, which is considered as one of the sole pointers to highlight the performance of each solution is calculated in Step 3. The recent literature of NIOA based MLT has been presented in Table 3. Different methods and parameter’s abbreviation used in the papers surveyed in Table 3 with its full form is tabularized respectively in Tables 4 and 5. Total 88 papers have been discussed in Table 3 where 19, 31, and 38 papers are collected from the years 2021, 2020 and 2019 respectively and presented in Fig. 3a. Whereas, Fig. 3b indicates the percentage of papers which are surveyed in Table 3 utilized different types of images.The mentioned papers have been collected from the following sources:

  1. (i)

    Google Scholar—http://scholar.google.com

  2. (ii)

    IEEE Xplore—http://ieeexplore.ieee.org

  3. (iii)

    ScienceDirect—http://www.sciencedirect.com

  4. (iv)

    SpringerLink—http://www.springerlink.com

  5. (v)

    DBLP—http://dblp.uni-trier.de

  6. (vi)

    ACM Digital Library—http://dl.acm.org

Table 4 Different method’s abbreviation used in the paper surveyed in Table 3 and its full form
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Table 5 Different qualitative parameters used in the paper surveyed in Table 3 and its full form
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Fig. 3
figure 3

Graphical analysis: a Year wise published papers; b number of papers surveyed using different types of images

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figure a

General approach of multi-level thresholding using NIOA

3.4 Major challenges of NIOA based multi-level thresholding

Based on the literature, it can be practically believed that NIOA has an enormous impact over MLT problem despite of some issues pertaining to few NIOA, particularly from theoretical perspectives as summarized in Sect. 2.1 that has clearly emphasized on some realistic challenges. In addition, the challenges specifically in regard to NIOA based MLT for image segmentation. Is thereby summarized as follows:

  1. 1.

    Selection of NIOA for MLT: It can be noticed that a massive set of NIOA has been introduced and exist in literature. Though, theoretically and practically, each NIOA’s performance extensively depends on the problem under consideration i.e. the image type for MLT. Nonetheless, it is not at all feasible to apply each NIOA over a set of images and assess its performance. Subsequently, for a researcher, appropriate selection of NIOA for a set of images turns out to be reasonably challenging. Current tradition also illustrates the application of newly developed NIOA over MLT and highlights a rigorous comparative study with other well-established NIOA in the problem domain.

  2. 2.

    Selection of Objective Function: Proper selection of objective function for a NIOA based MLT model for a specific set of images is also very demanding. Numerous objective functions are developed in the literature which makes the selection more crucial for a type of images like medical or satellite. For example, Paulo claimed that Tsallis’ entropy outperformed Cross and Shannon entropies by considering the segmentation of gray level images (Rodrigues et al. 2017). Ray et. al. (2021) claimed that Tsallis entropy is better than Otsu, Cross, Fuzzy entropy, Kapur’s entropy for proper thresholding of color pathology images. Whereas, Suresh et. al. claimed that Cross entropy is superior to Tsallis’ entropy to segment satellite images (Suresh and Lal 2017). Therefore, it can be said that the efficiency of the objective functions crucially depend on the NIOA and Image type.

  3. 3.

    Selection of Quality Parameters: As such there is no single criterion for completing accurate segmentation for different variants of images. Furthermore, there is no one metric for evaluating the segmentation quality of an image segmentation technique across various image types (Dhal et al. 2020a). Several quality parameters have been listed in Table 4 and some of their selection will judge which NIOA based MLT model is best for the set of image type under consideration.

  4. 4.

    Selection of Random walk for NIOA in the MLT domain: Randomization is a crucial part of any intelligence system, including current NIOA. Randomization allows an NIOA to leave any local optimum and search globally. Several random walks (Dhal et al. 2020b) demonstrated their effectiveness with various NIOA in various domains. Essentially, no analytical conclusions exist that proclaims which random walk is preferable for which algorithm. The application of any specific random walk, on the other hand, is entirely dependent on the problem and the NIOA in concern. As a result, choosing the right random walk during the development or improvement of NIOA is critical.

In addition to the above challenges, improvement of NIOA, their optimal parameters setting, their performance comparison in MLT based image segmentation domain are few other major challenging tasks that can be looked upon in future.

4 Experimental results

This section presents the experimental results which have been computed with the help of six NIOA and Masi entropy over satellite images. The six NIOA are Aquila Optimizer (AQO) (Abualigah et al. 2021a), Arithmetic Optimization Algorithm (AOA) (Abualigah et al. 2021b), Archimedes Optimization Algorithm (AROA) (Hashim et al. 2021), Rat Swarm Optimization Algorithm (RSA) (Dhiman et al. 2021), Particle Swarm Optimization (PSO) (Dhal et al. 2019c), and Firefly Algorithm (FA) (Dhal et al. 2020d). It can be noticed that four NIOA i.e., AQO, AOA, AROA, and RSA are developed in 2021 and they are very popular NIOA according to citation in Table 1. PSO and FA are well-established NIOA and selected to compare to these new NIOA for proving their effectiveness. Now, Masi entropy has been opted because it is state-of-the-art entropy and attracted many researchers in MLT domain over different kind of images. We maximize the Masi entropy for image segmentation. So, it is a maximization problem. For the reasonable comparison amongst NIOA methodologies, each execution of the tested objective functions considers the Number of Function Evaluations, NFE = 1000 × d, as stopping criterion of the optimization process. This criterion has been designated to encourage compatibility with previously published works in the literature. The experiments are evaluated considering the number of threshold values (TH) set to 5, 7, and 9which correspond to the d-dimensional search space in an optimization problem formulation. Furthermore, FE is also a crucial performance index used to measure the efficiency of NIOA. In comparison to computational complexity, FE permits some technical aspects such as the computer system where the experiments run and is implemented, that has direct impact on the running CPU time thereby concentrating only on the capacity of the algorithm to search within the solution space. For measuring the optimization ability of the NIOA, mean fitness \(\left(\overline{f }\right)\) and standard deviation \(\left(\sigma \right)\) have been calculated. On the other hand, the segmentation efficiency of the NIOA based models have been measured by computing Peak Signal-to-Noise Ratio (PSNR), Quality Index based on Local Variance (QILV), and Feature Similarity Index (FSIM). These parameters are very well-known in image segmentation domain. Table 4 highlights the list of utilized segmentation quality parameters (Dhal et al. 2020c, 2021b, 2021c; Das et al. 2021).

With the intention to verify the efficiency of different NIOA, experiment is conducted using 20 color satellite images. Further, MatlabR2018b and Windows-10 OS, × 64-based PC, Intel core i5 CPU with 8 GB RAM are the hardware and software requirements incorporated during the experiment. The proposed algorithms are tried and explored on images extracted from the site of Indian Space Research Organization (ISRO) (Dhal et al. 2019b) [https://bhuvan-app1.nrsc.gov.in/imagegallery/bhuvan.html#]. Figure 4 represents the original images of different satellite images. The parameter setting of the NIOA is given in Table 6.

Fig. 4
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Original satellite images

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Table 6 Parameter setting of the NIOA
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Figure 5 represents the segmented images of Fig. 4 by the tested NIOA. Table 7 signifies different parameters such as average numerical values of PSNR, QILV, FSIM, standard deviation \(\left({\sigma }_{f}\right)\), Computational time, and fitness function \(\left(\overline{f }\right)\) employed over satellite images to implement NIOAs with Masi entropy as an objective function. Average fitness and standard deviation clearly show that AROA provides the best results over thresholds 5, 7, and 9. PSO gives worst results among all the tested NIOAs. From Table 7, we can further conclude that if numbers of thresholds increase, value of PSNR, QILV, and FSIM also increase for the objective function. The fitness values of AROA are equated with other NIOAs using a nonparametric significance proof (Wilcoxon’s rank test) (García et al. 2009). Such proof permits evaluating differences in the outcome among two related methods. Wilcoxon's rank test at the 5% significance level is used to determine if the results attained with the best performing algorithm vary statistically significantly from the final results of the other competitors. A p-value of less than 0.05 (5% significance level) sturdily supports the condemnation of the null hypothesis, thereby signifying that the best algorithm's results vary statistically noteworthy from those of the other peer algorithms and that the discrepancy is not due to chance. Table 8 reports the p-values produced by Wilcoxon’s test for a pair-wise comparison of the fitness function between two groups formed as AROA vs. AOA, AROA vs. AQO, AROA vs. RSA, AROA vs. FA, and AROA vs. PSO for 5, 7, and 9 number of thresholds. All of the p-values in Table 8 are less than 0.05 (5% significance level), and h = 1 is clear proof against the null hypothesis, showing that the AROA fitness values for the performance are statistically higher, and this is not a fluke.

Fig. 5
figure 5

Segmented results of NIOA using Masi entropy over 5, 7, and 9 thresholds

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Table 7 Numerical comparison of NIOA for Masi entropy as objective function
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Table 8 Comparison among NIOA depending on Wilcoxon p-values
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5 Conclusion and future directions

This study has three major contribution as a survey paper which are (i) List of recent NIOA which felicitate the researchers for selecting and employing these new NIOA over different engineering optimization field, (ii) A survey table of NIOA based MLT for image segmentation which will give idea about the application of different NIOA with different objective functions over different kind of images, (iii) lastly a comparative study among some newly proposed NIOA have been performed with recently developed Masi entropy for satellite image multi-level thresholding. The experimental results are very encouraging due to the application of different NIOA over Masi entropy-based image segmentation.

Therefore, it can be easy to infer that NIOA based MLT is a fresh and exciting research topic with innovative methodologies. Applying diverse NIOA with various objective functions over several types of images is considered a complicated task. The main challenging future direction is the testing of this huge set of NIOA in MLT domain. Selection of proper objective function is also a difficult task because it is reported in literature that proper segmentation of specific kind of images is significantly depends on objective function. Exploration of multi-objective MLT can also be a great future work. Recently histogram clustering has been emerged as good alternative of histogram thresholding (Dhal et al. 2021c; Das et al. 2021). Therefore, application of NIOA for histogram based image clustering can also be a good future direction. Another application of NIOA will be for fuzzy rule classifier (Zhou and Angelov 2007; Angelov and Zhou 2008) or data analytics (Angelov et al. 2017).